Improvements of Lagrange's bound for polynomial roots. (September 2017)

• Record Type: Journal Article
• Authors: Batra, Prashant; Mignotte, Maurice; Ştefănescu, Doru
• Is Part Of: Journal of symbolic computation. Volume 82(2017)
• Page Start: 19
• Page End: 25
• Publication Date: 2017-09
• Subjects: Complex roots -- Overestimation factor -- Computational effort; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.10.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: An upper bound for the roots of X d + a 1 X d − 1 + ⋯ + a d − 1 X + a d is given by the sum of the largest two of the terms | a i | 1 / i . This bound by Lagrange has gained attention from different sides recently, while a succinct proof seems to be missing. We present a short, original proof of Lagrange's bound. Our approach leads to some definite improvements. To benefit computationally from these improvements, we construct a modified Lagrange bound which at the same asymptotic computational complexity is at most 11 per cent from optimal for degrees d ≥ 16 .